On the Three-Parameter Representation of the Equation of State Otto Redlich Inorganic Materiais Research Division Lawrence Berkeiey Laboratory. and Department of Chemicai Engineering University of Caiifornia Berkeiey California 94 720
Almost all small, nonpolar molecules satisfy the theorem of corresponding states; their P-V-T relation is quite well represented by a two-parameter equation propused in 1949. A third individual parameter is known to be required for long chains and polar molecules. The quality of the three-parameter representation has been examined by means of an equation of state based solely on the critical constants. The equation is reasonably convenient for practical applications, including the derivation of the fugacity coefficient. Mean deviations for 13 widely different substances confirm a fairly satisfactory algebraic representation by three individual parameters. A few exceptions, such as water, hydrogen and helium, are well known.
1. Introduction
T h e practical interest in a n equation of state rests preponderantly in the various thermodynamic properties t h a t are derived from it, especially the fugacity coefficient. It has been pointed out long ago t h a t we measure almost always the P-V-T relations but actually need the fugacity coefficient. For this reason and also for convenient use with computers a n adequate algebraic representation is desired. The problem seemed to be close to its solution 25 years ago when a simple equation of state was proposed (Redlich and Kwong, 1949). From the beginning it was manifest t h a t the “old equation” could not constitute a really satisfactory solution because it contained only two individual parameters; thus it was in accord with the theorem of corresponding states, which had been known to be deficient. In fact, the old equation was surprisingly satisfactory for a large group of small, nonpolar molecules, but it did not well represent long chains and polar molecules. Moreover, the work of Pitzer (1961) and his coworkers and of Riedel (1954-1956) showed t h a t three individual parameters are necessary and sufficient for a good representation of all substances with the exception of very few, such as hydrogen, helium and water. It was obvious to hope t h a t the reasonable introduction of a third parameter into a suitable modification of the old equation would solve the problem. Horvath’s review (1974) of the old equation contains 34 references to attempts a t improving its accuracy without too great a loss of simplicity and convenience. So far none of the results has been generally adopted. Perhaps it may be concluded that we should search more systematically for a new approach to this problem. Unquestionably the most difficult part is the representation of the critical point and its neighborhood. It occurred to us t h a t the most promising way might be to catch the bull by its horns, i.e., to start by constructing a three-parameter equation with the critical compressibility ratio as one parameter. The two other parameters are determined, as before, by the critical d a t a . One cannot expect, of course, t h a t a reasonably simple equation of this kind will result in sufficient accuracy. But one may hope to obtain a “main term” which reproduces approximately the peculiarities of the problem. The remaining discrepancies must then be eliminated as far as possible by additional terms. As a matter of course, these terms must not contain any new individual parameters. In the following only the reduced temperature, pressure and volume will be introduced; they will he denoted by T.
P and V. The critical temperature and pressure will not explicitly appear and only the critical compressibility factor will. Since most applications will he carried out by automatic computation, it will be convenient to make a concession to computer language in order to deal with the shortcoming of our usual language, namely, the lack of suitable symbols. We shall therefore write two (and occasionally even three) capital letters for a single quality. e.g., ZC for the critical compressibility factor. Multiplication will be indicated by a dot when necessary to avoid ambiguity. The choice of ZC as the third parameter is not seriously different from the use of Pitzer’s acentric factor o since there exists a fairly good relation Z C = 0.291 - 0 . 0 8 2 ~
(1)
between the two quantities. The primary basis of observed data was given by Pitzer’s tables, supplemented hy Lu and coworkers (1973) for the reduced temperatures 0.5 to 0.8. An array of 288 data was prepared according to the schedule: Reduced temperatures
Reduced pressures
0.5
0.2 0.4 0.6 0.8 1.o 1.2 1.6 2 .o 3 .O 5.0 7.0 9.0
0.6 0.7 0.8 0.9 1.o 1.1 1.2 1.6 2 .o 3 .O
4.O
Critical conipre s sibiliy factors 0.291 0.250
It will be seen that the “difficult” critical region is most closely covered; this should be taken into account in judging the magnitude ofdeviations. 2. T h e Main T e r m
For the development of the main term Z of the compressibility factor. one will of course follow the general guidelines that have been useful for the old equation. In other words. one concludes from Wegscheider’s discussion (1928) t h a t only a cubic equation with a reduced limiting volume B is acceptable. Thus the search for the term can Ind. Eng. Chem., Fundam.. Vol. 14, No. 3, 1975
257
Table I. Deviations in 2 Maximum deviation
Mean deviation
ZC
R and K
Main term
Final
R and K
Main term
Final
0.291 0.250
0.066 0.381
0.071
0.030 0.044
0.013 0.102
0.022 0.058
0.011 0.011
0.350
start from a n equation of the van der Waals type. The (reduced) relation
QQ
p . z c = -V -T- B
V2+FnV+G
Q
O =
- Q(3
+
3F
(4)
+
(5) (1 + (1 - B ) 3 we have written Q for the value of the function QQ a t the critical temperature. The conditions (3), (4), (5) are satisfied if the constants conform with the relations
+
G = B(3 - 2 B - l/ZG) Q = [I -
B
+ zc(1 -
zc(i -
(6) B)3
~ ) 3 3
(7) (8)
This leaves only the value B of the limiting volume open. In the old equation we had chosen the value 0.26, which is a fair average of individual values (Kuenen (1919) quotes values between 0.242 and 0.282). The satisfactory behavior of the old equation a t very high pressures is a consequence of the choice of the value of B Unfortunately the value 0.26 did not give good results in the equation ( 2 ) . A series of systematic tests led to the value
B = 0.352
(9)
which is significantly higher. But the difference is expected to be harmful only a t extreme pressures. For the representation of the temperature function QQ and other functions, we introduce a number of abbreviations. The symbols A T = 1, if T A T = 0, if T
